Universal minimal constants for polynomial growth of groups.

We study the minimal polynomial growth rate of finitely generated groups in the following sense. We prove that there exist positive numbers $\epsilon_d$ such that if $G$ is a group either of polynomial growth of degree $d$, or of non-polynomial growth, then that growth is at least $\epsilon_dn^d$. If $G$ is nilpotent, it suffices to assume that the degree is at least $d$. We indicate an application for random walks on groups.